Before diving into tips on using the exponent rules, we should start by answering one basic question: What is an exponent? In short, an exponent allows you to write math problems in short hand. Rather than writing 6 x 6 x 6 x 6 x 6, all you have to say is 65. In this instance, the 6 is known as the base and the 5 is the exponent or power. So, you would say 6 to the 5th power.
You can follow this language for other exponents. However, when you have a number to the second power you can say “squared,” such as 52 would be “five squared.” Also, any number to the third power would be “cubed”, such as 63 would be “six cubed.”
It might seem easy to solve math equations with exponents, especially once you memorize PEMDAS. However, it’s not always easy when working with variables. For those occasions, here are 8 exponent rules that you should know.
The One Property
If 68=6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 = 1,679,616, then 61= 6. If the exponent is 1, then you only have one of the base. So, the answer is your base number or variable. Knowing this, you can simplify the following equation:
On the flip side, if the base is 1 and the exponent is a variable, the equation can be simplified to one. Why? Because 1 x 1 = 1. It doesn’t matter how many times you multiply 1 by itself, you’re going to end up with one. Therefore, 1x =1.
The Zero Property
What does it mean when you have a number to the 0 power? How do exponent rules apply in this situation? As long as the number is a nonzero number, the equation is equal to 1.
This means that you’re going to multiply the number by itself zero times. To understand why this is the case, let’s take a look at an example. Let’s say that you have an equation like this:
From the One Property, we know that any number raised to the first power equals the number that you started with. So, the question is essentially, From the Negative Power Property, we learn that
And of course, 1-1=0. Therefore, 20=1.
To see how this works in action, consider the following equation:
The Negative Power Property
You may find an equation with a negative number as the exponent. This won’t work. What should you do then? Simply move it to the denominator. So, x-y becomes When working with numbers, you may have something like this 6 -2. Then, you would move the equation to the denominator and simplify it.
Check out this example:
In this case, you would switch the variables in order to make the exponents positive:
The Product Property
The Product Property is often referred to as the Identical Base Property, too. This is because you need to work with two variables that have the same base in order to use this property. Also, the base must be a nonzero real number. For example, 32 x 3 3 could be simplified by adding the exponents together: 32+3. In the end, you would have 35 or 243.
Now, let’s practice this property with variables. You might have (x4 y 5)( x2 y 6). Thanks to the Product Property, you can simplify this to x4+2 y 5+6 or x6 y 11.
The Quotient Property
As the Product Property relies on adding the exponents together, the Quotient Property calls on you to subtract the exponents. Similar to the Product Property, the bases must be the same and the variable can’t be 0. For example, would be If you simplify the problem, you end up with x2.
You can practice this property with the following equation:
First, you would think about the problem as 3x4-3. Then, simplify it to 3x.
The Power of a Power Property
What about when you come across a problem that looks like this: (xy)z? In this case, you have a power of a power. To solve this problem, you can multiply the exponents together. For example, (23)5 would become 23 x 5. From there, you would get 215 or 32,768.
The Power of a Product Property
The Power of a Product Property is similar to the Power of a Power Property. However, you’re going to be working with 2 bases in the parentheses instead. For this property, the exponent should be applied to both bases. For example, (xy)z would become xzyz. Check out the following example to see how this could be applied in an equation:
22 x 2x4 x 2y6 x 2
The Power of a Quotient Property
When dealing with quotients, you may find yourself with an exponent on the outside of the parentheses. This means that, like the Power of a Product Property, you need to apply the exponent to each base within the parentheses. For example, would become .
Follow along with this equation:
When you come across an equation with exponents, think about each of the exponent rules. Start by identifying which of the exponent rules you need to use to solve the equation. Then, you can quickly determine the best way to simplify the problem and reach the given solution. Be sure to follow the order of operations to follow the correct steps for the equation. Then, simplify to discover the correct solution.